Non-Abelian Josephson effect Non-Abelian Josephson effect and fractionalized vorticesand fractionalized vortices

Wu-Ming Liu（刘伍明）（ Institute of Physics, CAS）

Email: wmliu@aphy.iphy.ac.cn

Supported by NSFC, MOST, CAS

CollaboratorsCollaboratorsJiang-Ping Hu (Purdue Univ)An-Chun JiZhi-Bing Li (Zhongshan Univ)Ran QiQing SunXin-Cheng Xie (Oklahoma State Univ)Xiao-Lu YuYan-Yang ZhangFei Zhou (British Columbia Univ)

1. Introduction2. Non-Abelian Josephson effect3. Josephson effect of photons4. Localization5. Fractionalized vortex6. Outlook

Outline

1.1. BEC of ideal gas 7Li 6Li

1. Introduction1. Introduction

1.2. BEC in dilute gas1.2. BEC in dilute gas

1.3. BEC near Feshbach resonance1.3. BEC near Feshbach resonance

1.4. BEC in optical lattices1.4. BEC in optical lattices

1.5. Fermionic condensation

1.6. Molecule condensation?J.G. Danzl et al. Science 321, 1062 (2008)

R. Qi, X.L. Yu, Z.B. Li, W.M. Liu,

Non-Abelian Josephson effect between two F=2 spinor Bose-Einstein

condensates in double optical traps,

Phys. Rev. Lett. 102, 185301 (2009)

2. Non-Abelian Jesephson effect2. Non-Abelian Jesephson effect

Abelian case:U(1) × U(1) U(1) diagonaltwo goldstone modes one gapless

mode (goldstone mode) and one gapped mode (pseudo goldstone mode)

Non-Abelian case:Non-Abelian case:SO(N), U(1) SO(N), U(1) × SO(N)…SO(N)…Multiple Multiple pseudo goldstone modes

No Josephson effect

U(1)XU(1)Nambu-Goldstone modes

Josephson effect

Single mode:U(1)XU(1)Nambu-Goldstone modesMany modes:S=1, U(1)XS(2);S=2, U(1)XSO(3)Pseudo Nambu-Goldstone modes

Ground states of S=2 boson

Ferromagnetic phaseAntiferromagnetic phaseCyclic phase

Ferromagnetic phase

U(1)XU(1)Nambu-Goldstone modes

Antiferromagnetic phase

U(1)XSO(3)Pseudo Nambu-Goldstone modes

Cyclic phase

U(1)XSO(3)Pseudo Nambu-Goldstone modes

Antiferromagnetic phase

m=0

m=±2

Fig. 2 The frequencies of pseudo Goldstone modes as a function of coupling parameter J in the case of antiferromagnetic phase.

Cyclic phasem=±1

m=0,±2

Fig. 3 The frequencies of pseudo Goldstone modes as a function of coupling parameter J in the case of cyclic phase.

Experimental parameter

Rb-87, F=2AFM: c2<0, c1-c2/20>0Cyclic: c1>0, c2>0c1:0-10nK, c2:0-0.2nK, c0:150nKfluctuation time scale-10mspseudo Goldstone modes:1-10nk

Experimental signatures

Initiate a density oscillationDetect time dependence of atom numbers in different spin component◆Measure density oscillation in each of spin componentsNon-Abelian Josephson effect

A.C. Ji, Q. Sun, X. C. Xie, W. M. Liu,

Josephson effects of photons in two weakly-inked microcavities,

Phys. Rev. Lett. 102, 023602 (2009)

3. Jesephson effect of photons3. Jesephson effect of photons

Fig. 1 Experimental setup and control of coupling along resonator axis

Fig. 2 Excitations of a polariton condensate

Fig. 3 Chemical potential-current relation in polariton condensates

4. Localization4. LocalizationJ. Billy et al., Nature 453, 891 (2008).J. Billy et al., Nature 453, 891 (2008).

G. Roati et al., Nature 453, 895 (2008)

Y.Y. Zhang, J.P. Hu, B.A. Bernevig, X.R. Wang, X.C. Xie, W.M. Liu,

Localization and Kosterlitz-Thouless transition in disordered graphene,

Phys. Rev. Lett. 102, 106401 (2009)

ABAA

B

B

Fig. 1 The scaling function

Fig. 2 Typical configurations of local currents In (red arrows)and potential Vn (color contour) on two sides of K-T type MIT with N=56X32 sites, \xi=1:73a, nI=1% and EF=0:1t. (a) W=1:1t (delocalized); (b) W=2:9t (localized).

A.C. Ji, W.M. Liu, J.L. Song, F. Zhou,

Dynamical creation of fractionalized vortices and vortex lattices,

Phys. Rev. Lett. 101, 010402 (2008)

5. Half vortex5. Half vortex

Dynamical creation of fractionalized vortices and vortex lattices

Fig.1 Density and spin density of an individual half vortex

Fig. 2 Interaction potentials between two half vortex

220

0

_

2 1 1 0 2 1 1 0

( ) [2

( )] 2

tr zi V L ct m

c c

hh

221

_2

0 2 1 0 1 1 2 0 1

( ) [2

( ) ]

tr zi V Lt m

c c W c

m m

hh m

2

2 21 12

2 250 Hz

trmV r x y

2

2 2

2mW r x y

m

Fig. 3 Creation of a half-quantum vortex. The bottom panel shows that a single half vortex is formed at t=600 ms after magnetic trap has been adiabatically switched off.

(a) Creation of a triangular integer vortex lattice

(b) A square half vortex lattice formation at t=1600 ms

6. Outlook6. Outlook

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